improved orientation sampling for indexing diffraction ... · in many types of diffraction...
TRANSCRIPT
General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
You may not further distribute the material or use it for any profit-making activity or commercial gain
You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Downloaded from orbit.dtu.dk on: Nov 06, 2020
Improved orientation sampling for indexing diffraction patterns of polycrystallinematerials
Larsen, Peter Mahler; Schmidt, Søren
Published in:Journal of Applied Crystallography
Link to article, DOI:10.1107/S1600576717012882
Publication date:2017
Document VersionPublisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):Larsen, P. M., & Schmidt, S. (2017). Improved orientation sampling for indexing diffraction patterns ofpolycrystalline materials. Journal of Applied Crystallography, 50(6). https://doi.org/10.1107/S1600576717012882
research papers
J. Appl. Cryst. (2017). 50 https://doi.org/10.1107/S1600576717012882 1 of 12
Received 12 July 2017
Accepted 8 September 2017
Edited by Th. Proffen, Oak Ridge National
Laboratory, USA
Keywords: orientation sampling; pattern
indexing; forward modelling; quaternions;
spherical coverings.
Improved orientation sampling for indexingdiffraction patterns of polycrystalline materials
Peter Mahler Larsen* and Søren Schmidt
Department of Physics, Technical University of Denmark, 2800 Kongens Lyngby, Denmark. *Correspondence e-mail:
Orientation mapping is a widely used technique for revealing the microstructure
of a polycrystalline sample. The crystalline orientation at each point in the
sample is determined by analysis of the diffraction pattern, a process known as
pattern indexing. A recent development in pattern indexing is the use of a brute-
force approach, whereby diffraction patterns are simulated for a large number
of crystalline orientations and compared against the experimentally observed
diffraction pattern in order to determine the most likely orientation. Whilst this
method can robustly identify orientations in the presence of noise, it has very
high computational requirements. In this article, the computational burden is
reduced by developing a method for nearly optimal sampling of orientations. By
using the quaternion representation of orientations, it is shown that the optimal
sampling problem is equivalent to that of optimally distributing points on a four-
dimensional sphere. In doing so, the number of orientation samples needed to
achieve a desired indexing accuracy is significantly reduced. Orientation sets at a
range of sizes are generated in this way for all Laue groups and are made
available online for easy use.
1. Introduction
In many types of diffraction experiments, the aim is to
determine the orientation of the diffracted crystallite volume
which creates the experimentally observed pattern. For
example, when studying a multigrain sample with the three-
dimensional X-ray diffraction technique (3DXRD; Poulsen et
al., 2001), a ‘grain map’ is constructed by finding the crystalline
orientation at each point in the sample. The process of
determining the crystalline orientation from a diffraction
pattern on the detector is known as pattern indexing.
Throughout this article, we use the term ‘experimental
pattern’ to denote an image of a diffraction pattern as
recorded on a detector.
The most widely used pattern indexing methods work
‘backwards’ from features in the observed data to an orien-
tation. Such methods are typically highly efficient, but can fail
in the presence of noise. A well known example is in electron
backscatter diffraction (EBSD), where the Hough transform is
used to find lines in the backscattered Kikuchi pattern, from
which the orientation can be determined (Adams et al., 1993).
Under noisy conditions, however, the Kichuchi lines can no
longer be reliably identified and the indexing process fails as a
consequence.
The desire to analyse diffraction patterns under less-than-
ideal conditions has motivated the development of forward-
modelling-based pattern indexing, also known as dictionary-
based indexing. In a forward model, rather than working
backwards from the data, the orientation is found using a
brute-force approach. A dictionary is constructed by selecting
ISSN 1600-5767
# 2017 International Union of Crystallography
a set of orientations and generating simulated patterns for
each of them. A requirement for simulating patterns is that the
crystal phase is known a priori, or, if indexing a multiphase
material, that the set of candidate phases is known.
To index an experimental pattern, it is compared against
every simulated pattern in the dictionary, and the dictionary
pattern with the highest similarity determines the orientation
(in a multiphase material this also determines the phase).
Here, the similarity is determined by the difference in the pixel
intensities in the simulated and experimental patterns. By
using the full image information (i.e. all pixel intensities), the
similarity exhibits a continuous degradation with increasing
noise, as opposed to the catastrophic degradation exhibited
when looking for specific features in the experimental pattern.
A significant drawback of the forward-modelling approach,
however, is the computational effort required: each experi-
mental pattern must be tested against every dictionary
pattern. Since, the accuracy of the pattern indexing process
depends on the granularity of the set of dictionary orienta-
tions, a more accurate indexing requires a larger set.
Increasing the number of dictionary orientations, however,
increases the time required to index a pattern. Since the
objectives of increased accuracy and reduced running time are
in opposition to each other, we ask the question, how can we
achieve the highest accuracy with the fewest dictionary
orientations? In this article, we describe a method for doing so
with the use of quaternions.
1.1. Measurement of dictionary orientation sets
Orientations can be conveniently represented using unit
quaternions (Altmann, 2005). Briefly, a quaternion is a four-
dimensional vector of the form q ¼ fw; ix; jy; kzg, where w, x,
y and z are real numbers and i, j and k are imaginary numbers
which generalize the better-known complex numbers. Unit
quaternions represent points on a four-dimensional hyper-
sphere, a space formally known as S3 and which consists of all
vectors that satisfy ðw2 þ x2 þ y2 þ z2Þ1=2¼ 1. This space is a
double covering of SOð3Þ, the group of rotations in three-
dimensional Euclidean space (R3). The double covering
relationship means that �q and q represent the same orien-
tation, which is evident when considering the quaternion-
derived rotation matrix:
Uq ¼
1� 2y2 � 2z2 2xy� 2wz 2xzþ 2wy
2xyþ 2wz 1� 2x2 � 2z2 2yz� 2wx
2xz� 2wy 2yzþ 2wx 1� 2x2 � 2y2
24
35: ð1Þ
It can be seen that in each element of Uq the sign of the
quaternion cancels out. By using the quaternion representa-
tion, the problem of selecting an optimal set of dictionary
orientations is equivalent to finding an optimal distribution of
a set of points on S3. To do so, we must first decide what
constitutes a good distribution.
The misorientation between two orientations in quaternion
form, p and q, is given by
�ðp; qÞ ¼ 2 arccos jhp; qij; ð2Þ
where hp; qi denotes the inner product of p and q. In many
previous studies, dictionary orientation sets are quantified by
the misorientation between neighbouring orientations: for
example, the average value of �ðp; qÞ over all pairs of nearest
neighbours p and q. This may be adequate when the orien-
tation set has a known grid-like structure, but it does not
constitute a universal measure of quality. To illustrate this with
a pathological example, consider an orientation set, Q, where
all orientations lie at the same point. The misorientation
between all pairs of orientations is zero, that is
�ðp; qÞ ¼ 0 8p 2 Q; q 2 Q; ð3Þ
yet the set constitutes the worst possible dictionary. A good
measure of quality should instead consider the misorientation
between the dictionary set and any possible experimental
orientation. We define the error term as the maximum
misorientation between these two, i.e. how far can an experi-
mental orientation lie from the dictionary? More specifically,
this error term is given by
�max ¼ maxx2SOð3Þ
minq2Q
� x; qð Þ: ð4Þ
This quantity can be minimized by solving the spherical
covering problem in S3. Given N hyperspherical caps of equal
radius, r, called the covering radius, the spherical covering
problem asks how to arrange the caps to cover the surface of
S3 with minimal r. We describe this problem in detail in x2.
By creating orientation sets with a small covering radius, we
can either reduce the number of orientations required to
achieve a desired error tolerance (thereby reducing the
running time of forward-modelling pattern indexing) or
simply improve the error distribution for a fixed number of
orientations. Creation of such sets is the principal contribution
of this work.
1.2. Previous work
Forward modelling has been successfully applied in many
types of diffraction-based experiments, including the indexing
of three-dimensional X-ray diffraction microscopy data (Li &
Suter, 2013; Schmidt, 2014), EBSD data (Chen et al., 2015) and
electron channelling patterns (Singh & De Graef, 2017). Any
forward-modelling method requires a discretization of SOð3Þ.
Whilst many such discretization methods have been devel-
oped, here we consider only three which are both successful
and commonly used amongst crystallographers.
Yershova et al. (2010) have developed an incremental infi-
nite sequence based on the Hopf fibration. The method
generates orientations deterministically, with proven maximal
dispersion reduction when used as a sequence. Furthermore,
the orientation sets are isolatitudinal, which permits expansion
into spherical harmonics (Dahms & Bunge, 1989), refinable
and can be generated on the fly. Whilst the method has many
desirable properties, it is developed for the purpose of robot-
motion planning and is not easily integrated with crystal-
lographic fundamental zones. To remedy this, Rosca et
al.(2014) have developed ‘cubochoric’ coordinates, in which
an area-preserving Lambert projection is used to map points
research papers
2 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50
from a cubic grid onto any desired crystallographic funda-
mental zone in SOð3Þ. A different approach, developed by
Karney (2007) for use in molecular modelling, is to generate
sets which attempt to solve the spherical covering problem.
Inspired by the observation that body-centred cubic (b.c.c.)
grids solve the covering problem in R3, b.c.c. grids are
constructed in Rodrigues–Frank (RF) space (Frank, 1988;
Morawiec & Field, 1996) in order to generate good coverings
in SOð3Þ.
Table 1 summarizes the properties of the different methods
of generating orientation sets. Each of the three aforemen-
tioned techniques attempts to solve slightly different problems
and involves different trade-offs as a consequence, although
one feature they have in common is fast generation. We take
an alternative approach, sacrificing other properties in pursuit
of creating the ‘best’ possible orientation sets. Whilst this
approach requires a significant up-front computational effort,
this is a good trade-off when the resulting sets will subse-
quently be used many times. We emphasize that, whilst the
orientational error is critical to forward modelling, there are
many other sources of error in any modality [see Ram et al.
(2017) for a comprehensive analysis in an EBSD context].
The rest of this article is organized as follows: in x2 we
define the spherical covering problem on Sd, show how this
relates to the problem of finding an optimal set of orientations
and derive a conjectured lower bound. We describe the
generation of orientation sets in x3. Results on the covering
radius and error distributions of the resulting orientation are
given in x4. Lastly, the advantages and drawbacks of the
method presented are discussed in x5.
2. Error quantification of orientation sets
In order to compare different orientation sets we must define a
measure of quality. Here, we describe the covering radius of a
set, which we argue is the canonical error measure since it
determines the maximum possible error. We will first describe
the sphere covering problem for Euclidean and spherical
geometries, and then show that the problem of generating
optimal orientation sets is a special case of the spherical
covering problem.
2.1. Spherical coverings
The sphere covering problem is best known in Euclidean
geometries. In Rd, it asks ‘for the most economical way to
cover d-dimensional space with equal overlapping spheres’
(Conway & Sloane, 1998). Optimal coverings are known for
d ¼ 1 and d ¼ 2, which are equally spaced points on a line and
a hexagonal lattice, respectively, and optimal lattice coverings
are known for 1 � d � 5.
The presence of curvature in spherical geometries renders
the covering problem vastly more challenging. In Sd, the
spherical covering problem asks for the most economical way
to cover the surface of Sd with equal overlapping hyper-
spherical caps. In S1, the optimal covering is a set of N points
with angle 2�=N between adjacent points. For d> 1, however,
there is no general formula for determining the optimal
spherical covering. Furthermore, unlike in Rd, the configura-
tion of the optimal covering depends on the number of points
in the covering. For example, for d ¼ 2, the known optimal
configurations are the vertices of the tetrahedron, the octa-
hedron and the icosahedron. Hardin et al. (2017) have found
putatively optimal coverings for d ¼ 2 at other values of N,
but these have been found using numerical optimization and
are not provably optimal.
2.2. Covering radius and covering density
For coverings on Sd, the two (equivalent) measures of
quality are the covering radius and the covering density. Given
a discrete collection of points P ¼ fp1; p2; . . . ; pNg 2 Sd, the
covering radius, �, is defined as the largest angular distance
between any point in Sd and P, that is
� ¼ maxx2Sd
minp2P
arccoshx; pi; ð5Þ
where hx; pi denotes the inner product of x and p. Then, P
covers the surface of Sd with N ¼ jPj equal hyperspherical
caps of radius �. The covering density, �dð�Þ, is given by the
ratio of the sum of the surface area of the caps to the surface
area of a unit d-sphere,
�dð�Þ ¼ NCdð�Þ
Sdð1Þ; ð6Þ
where
Cdð�Þ ¼
Ztanð�Þ
0
Sd�1ðrÞ
ð1þ r2Þ2 dr; Sd�1ð�Þ ¼
d�d=2
� d=2þ 1ð Þ�d�1: ð7Þ
Sd�1ð�Þ is the surface area of the d-sphere of radius � and Cdð�Þis the surface area of a hyperspherical cap of radius � (cf.
Appendix A for derivation). To find the covering radius, we
need to determine the Voronoi cell of each point pi 2 P. The
Voronoi cell of point pi, denoted VorðpiÞ, consists of all points
of Sd that are at least as close to pi as to any other pj. More
specifically
research papers
J. Appl. Cryst. (2017). 50 Larsen and Schmidt � Improved orientation sampling for indexing 3 of 12
Table 1Summary of properties of different methods of orientation setgeneration.
Existing methods prioritize fast generation and a grid-like structure. In ourwork we optimize the covering radius at the expense of all other properties.The optimality gap for a set of N orientations is the percentage difference of itscovering radius to that of the simplex bound (cf. x2.5)
Method
Fastgenera-tion
Refin-able
Isolati-tudinal
Iso-choric
Crystallo-graphic
Optimalgap atN ’ 105 (%)
Random samplingp p
–† –p
127Hopf fibration
p p p– – 59.9
Cubochoricp p p p p
40.8Octahedral b.c.c.
p p–† – –‡ 15.4
Present work – – –† –p
4.64
† Non-isolatitudinal sets do not permit an expansion into spherical harmonics, thoughany orientation set can be expanded into hyperspherical harmonics (Mason & Schuh,2008; Mason, 2009). ‡ These orientation sets can be mapped out into seven of 11 Lauegroup fundamental zones (cf. x2.4).
VorðpiÞ ¼ fx 2 Sdj arccoshx; pii � arccoshx; pji 8jg: ð8Þ
Since the vertices of the Voronoi cells are the points which
locally maximize the angular distance from P, the covering
radius is determined by the Voronoi vertex that lies furthest
from P.
The Voronoi cells of a set of points in Sd are not easy to
calculate directly, so instead we calculate the (hyperspherically
constrained) Delaunay triangulation. The Delaunay triangu-
lation, DTðPÞ, is a set of hyperspherical simplices whose
vertices are points in P which satisfy the empty-sphere
condition, that is, no points in P lie inside the circumhypercap
of any simplex in DTðPÞ. Associated with each simplex is a
Voronoi vertex, which lies at the centre of the simplex
circumhypercap (the circumcentre). The Delaunay triangula-
tion, Voronoi cells, and simplex circumhypercaps and
circumcentres are illustrated in S2 in Fig. 1. We now show how
to calculate the circumcentre of a simplex.
Theorem. For a hyperspherical simplex t 2 DTðPÞ with
vertices fp1; p2; p3; . . . ; pdþ1g 2 Sd, the position of the
circumcentre, X, is equal to the unit normal vector of the
d-dimensional hyperplane on which the vertices of t lie.
Proof. Let S ¼ fs1 ¼ p2 � p1; s2 ¼ p3 � p1; s3 ¼ p4 � p1;. . . ; sd ¼ pdþ1 � p1g and let X 2 Sd be the circumcentre of t.
Then, per definition, X must satisfy
piX ¼ p1X 8i: ð9Þ
Subtracting p1X from each side gives
siX ¼ 0 8i: ð10Þ
The unit length of X follows from requiring X 2 Sd:
To find X, we calculate the normalized d-fold vector cross
product (Brown & Gray, 1967) of S. Since every hyperplane
has two (opposite) unit plane normals, X has two solutions,
which correspond to the centre of the simplex hypercircumcap
and its dual. However, given that jPj � dþ 2 only one of these
solutions fulfils the empty-sphere condition, which is the one
which satisfies hX; pii> 0 8i. This corresponds to the smaller
of the two hypercircumcaps.
For a set of points on Sd, the vertices of each simplex
t 2 DTðPÞ can be found by calculating the convex hull of P, as
shown in Fig. 2. If we denote the circumradius of a simplex t by
’ðtÞ, equation (5) can be restated as
� ¼ maxf’ðtÞ j t 2 DTðPÞg; ð11Þ
which provides a practical solution to equation (5): the
covering radius of a point set is simply the maximum simplex
circumradius.
2.3. Orientation sets
The problem of finding a good spherical covering is
immediately relatable to the problem of finding good sets of
orientations. As described previously, rotations can be repre-
sented by quaternions, which are points on S3. The maximum
rotational angle between a point x 2 SOð3Þ and a point set P,
also called the maximum misorientation, is given by
�max ¼ 2 maxx2S3
minP2P
min arccosh�x; pi; arccoshx; pi½ �;
¼ 2 maxx2S3
minp2Q
arccoshx; pi;ð12Þ
where Q ¼ P [ f�p j p 2 Pg. It can be seen that, for a point
set with antipodal symmetry, �max ¼ 2�, that is, the maximum
misorientation is twice the covering radius. Thus, the problem
of finding a set of rotations with the lowest maximum misor-
ientation is equivalent to finding an optimal spherical covering
for a point set with antipodal symmetry on S3.
2.4. Integration with crystallographic symmetries
Equation (12) shows that a set of 2N points with antipodal
symmetry represents a set of N rotations. A set of orientations
generated in this way covers the whole space of SOð3Þ and is
immediately applicable to pattern indexing of materials with
triclinic (C1) Bravais lattices. For materials with higher-order
symmetry, though, a dictionary set which covers all of SOð3Þ is
research papers
4 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50
Figure 2Convex hull (left) and the spherically constrained Delaunay triangulation(right) of 22 points on the sphere. The triangulations exist in R3 and S2,respectively, but the vertices of each simplex are the same. Data fromHardin et al. (2017).
Figure 1Left: a putatively optimal spherical covering for 28 points in S2 [point setfrom Hardin et al. (2017)]. The solid lines indicate the sphericallyconstrained Delaunay triangulation. The dashed lines indicate theVoronoi cells. Right: the same points; each simplex in the Delaunaytriangulation has a circumcap, the centre of which (marked in red) lies ata Voronoi cell vertex. The maximum simplex circumradius determines thecovering radius of the point set.
wasteful, since only the fundamental zone orientations (He &
Jonas, 2007) are needed. A naive approach for selecting
fundamental zone orientations is to generate a full covering of
SOð3Þ and then simply ‘cut out’ the desired region; this
introduces artifacts at the boundaries of the fundamental zone
which increase the covering radius significantly. Instead, we
apply the symmetry of the desired point group during
generation of the orientation sets.
Given a set of basis points B ¼ fb1; b2; . . .g and a quater-
nion group G ¼ fg1; g2; . . .g, we can create a set of points with
the symmetry of G by
P ¼ fb� g j b 2 B; g 2 Gg; ð13Þ
where � denotes quaternion multiplication. If P is to repre-
sent a set of orientations [cf. equation (12)], G must be a
superset of antipodal symmetry (C1). The finite quaternion
groups which meet this requirement are (Conway & Smith,
2003)
2I60 The binary icosahedral group
2O24 The binary octahedral group
2T12 The binary tetrahedral group
2Dn The binary dihedral group
2Cn The binary cyclic group
With the exception of the binary icosahedral group, each of
these is used to describe the generators of the 11 Laue groups
(Morawiec, 2004), C1, C2, C3, C4, C6, D2, D3, D4, D6, T and O.
By the application of a symmetry group, the problem of
finding a good spherical covering for a chosen crystallographic
fundamental zone is reduced to a problem of finding an
optimal configuration of the basis points, which is a much
smaller problem.
The Laue groups can be divided into two sets:
fC2;C4;D2;D4;T;Og ð14Þ
and
fC3;C6;D3;D6g; ð15Þ
where the elements of each are subsets of O and D6, respec-
tively (C1 is trivially a subset of both). This means that, if we
generate sphere coverings with O and D6 applied according to
equation (13), then by an appropriate mapping of the funda-
mental zone orientations we obtain sphere coverings for all
Laue groups, without the aforementioned boundary artifacts.
The Laue group subset relationships are shown in Appendix B.
2.5. Derivation of the simplex bound on S3
In addition to knowing the covering radius and density of a
point set, it is useful to know how far from optimality a set is.
We can estimate the optimality gap with a lower bound.
The simplex bound is a classic result which gives an upper
bound on the density of sphere packings and a lower bound on
the density of sphere coverings. It has been proven for pack-
ings in Rd (Rogers, 1958) and Sd (Boroczky, 1978), and for
coverings in Rd (Coxeter et al., 1959) and S2 (Toth, 1964).
Boroczky has conjectured that it is a lower bound on S3
(Boroczky, 2004). Despite lacking a proof, we will use the
simplex bound on S3 to estimate the optimality of our point
sets, as it is ‘intuitively obvious’.
The premise of the simplex covering bound is that the
lowest covering density can be achieved with regular
simplices; this concept is illustrated in Fig. 3. Regular simplices
tesselate in R1 and R2. In Rd for d � 3, regular simplices do
not tesselate, and thus the simplex covering density is an
unattainable lower bound. As stated previously, regular
simplices tesselate in S2 for three configurations: the tetra-
hedron, the octahedron and the icosahedron. Thus, the
simplex bound is tight for these configurations only, and is
provably unattainable for any other number of vertices. In S3,
regular simplices tesselate only in the 5-cell, the 16-cell and the
600-cell. If Boroczky’s conjecture is correct, the simplex bound
is tight only for these configurations. Since no description of
the simplex bound covering density on S3 could be found in
the literature, we derive an expression for it here.
Consider a hyperspherical cap on S3 of radius � and volume
C3ð�Þ. We denote the inscribed regular spherical tetrahedron
Tð�Þ. At each of the four vertices of Tð�Þ is a hyperspherical
cap of radius �. Each of these caps intersects Tð�Þ with solid
angle �ð�Þ, giving a volume of intersection of C3ð�Þ�ð�Þ=ð4�Þ.Now Tð�Þ is covered by the four equal volumes of intersection.
The covering density, �S
3 , is the ratio of the sum of the four
volumes of intersection to the volume of Tð�Þ:
�S
3 ð�Þ ¼ 4C3ð�Þ�ð�Þ
4�
1
Vol½Tð�Þ�: ð16Þ
Here,
C3ð�Þ ¼ �½2� � sinð2�Þ�; ð17Þ
�ð�Þ ¼ 3 ð�Þ � �; ð18Þ
ð�Þ ¼ arccos4 cos2ð�Þ � 1
8 cos2ð�Þ þ 1
� �; ð19Þ
Vol½Tð�Þ� ¼ �ReðLÞ þ �½argð�QÞ þ 3 ð�Þ� � 32�
2� �
mod 2�2;
ð20Þ
research papers
J. Appl. Cryst. (2017). 50 Larsen and Schmidt � Improved orientation sampling for indexing 5 of 12
Figure 3Illustration of the simplex bound in R2, shown here because of thedifficulty of visualizing the simplex bound in S3. Regular simplices in R2
are equilateral triangles, which tessellate. At the vertices of each triangle(of circumradius r) is a circle of radius r. The area of intersection betweena triangle and a circle is a circular sector of angle �=3. Each triangle iscovered by three equal areas of intersection. The covering density istherefore the ratio of the sum of the three areas of intersection to the areaof the triangle: �
R2 ¼ 2�=3ð3Þ1=2. In R
d the covering density isindependent of r, which is not the case in Sd for d � 2 owing to a lackof tesselation.
Q ¼ 3 exp½�2i ð�Þ� þ 4 exp½�3i ð�Þ� þ exp½�6i ð�Þ�; ð21Þ
L ¼ 12
�Li2ðZ0Þ þ 3Li2fZ0 exp½�4i ð�Þ�g
� 4Li2f�Z0 exp½�3i ð�Þ�g � 3 ð�Þ2�; ð22Þ
Z0 ¼�6 sin2
½ ð�Þ�
Qþ
2�fcos½ ð�Þ� þ 1g3f1� 3 cos½ ð�Þ�g
1=2
Q:
ð23Þ
where ð�Þ is the dihedral angle of Tð�Þ. The terms in equa-
tions (17)–(19) are derived in Appendix A. Equations (20)–
(23) are a simplification of Murakami’s formula for the volume
of a spherical tetrahedron (Murakami, 2012), for the case
where all six dihedral angles are equal (a regular spherical
tetrahedron).
The covering density can be used to estimate the optimality
gap of a point set. For a set of N points with covering radius �,the lower bound on the covering radius �� can be found by
rearranging the density expression in equation (6):
N ¼2�2�
S3 ð��Þ
C3ð��Þ; ð24Þ
where 2�2 is the surface area of S3. The optimality gap of the
point set is then �=�� � 1. Since �S
3 ð��Þ is a nontrivial
expression, we find �� numerically.
3. Method of orientation set generation
We now describe the method for generating point sets with
small covering radii. The direct problem formulation with the
application of symmetry is shown in Table 2. This is essentially
just a restatement of equations (11) and (13).
The problem of finding optimal spherical coverings is
difficult; in addition to being an NP-hard problem (van Emde
Boas, 1981), the objective function is non-differentiable, and
the ‘fitness landscape’ is non-convex and has many local
minima. One possible solution approach [used by Hardin et al.
(2017) to generate coverings in S2] is to use direct search. This
overcomes the non-differentiability of the objective function,
but repeated solution from many different starting config-
urations is required to find the globally optimal configuration.
Furthermore, owing to the poor scaling of direct search
methods with increasing problem size, this approach is not
practical since we wish to create very large orientation sets.
Since it is unlikely that we will find globally optimal solu-
tions for large point sets with direct search, we will instead
attempt to find good solutions with an indirect method. We
proceed as follows: an initial set of orientations is created by
sampling randomly from a uniform distribution on SOð3Þ
(Shoemake, 1992). The covering radius is then succesively
reduced, firstly by using gradient descent to find a configura-
tion which is a local minimizer of the Riesz energy. Secondly, a
smoothing procedure is used to improve the characteristics of
the Delaunay triangulation. Lastly, a local optimization
procedure is used to further refine the solution. We present no
theoretical basis for the choice of methods or for the order in
which the methods are applied. Rather, empirical experi-
mentation has shown that the method is effective and
produces point sets with a small covering radius.
The motivation for choosing these methods is illustrated in
Fig. 4. The random point set has a large covering radius. By
minimizing the Riesz energy the covering radius is signifi-
cantly reduced. The covering radius can be further reduced as
shown in the optimal covering. The effect of the smoothing
procedure is not shown here, as it is visually very similar to the
Riesz energy and optimal covering configurations. In the rest
of this section we describe each method in detail.
3.1. Riesz energy minimization
For a set of points P ¼ fp1; p2; . . . ;pNg 2 S
d, the Riesz energy is defined as
EsðPÞ ¼
XN
i6¼j
1
jpi � pjjs if s> 0;
XN
i6¼j
log1
jpi � pjjif s ¼ 0:
8>>>><>>>>:
ð25Þ
The problem of finding optimal Riesz
energy configurations is well studied,
most commonly for ðd ¼ 3; s ¼ 1Þ
(also known as the Thomson problem)
(Erber & Hockney, 1991; Altschuler et
research papers
6 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50
Table 2Direct model for minimizing the covering radius of a point set in S3.
The point set P is composed of a basis set, B, on which a chosen symmetrygroup, G, acts. The covering radius, �, is calculated using the Delaunaytriangulation of P.
Variables B ¼ fb1 2 S3; b2 2 S
3; . . .gParameters G ¼ fg1 2 S
3; g2 2 S3; . . .g
Minimize � ¼ maxf’ðtÞ j t 2 DTðPÞgSubject to P ¼ fb� g j b 2 B; g 2 Gg
Figure 4Delaunay triangulations and Voronoi cells of three different point sets in S2, for N ¼ 130. Left: pointssampled uniformly from a random distribution. Centre: the global minimum configuration for theRiesz energy, here for s ¼ 1 [point set from Wales & Ulker (2006)]. Right: putatively optimalspherical-covering configuration [point set from Hardin et al. (2017)]. Point sets in S2 are used herefor illustrative purposes only, because of the difficulty of visualizing S3.
al., 1994; Wales & Ulker, 2006), but also for ðd ¼ 4; s ¼ 1Þ
(Altschuler & Perez-Garrido, 2007), and in the general case
(Hardin & Saff, 2004; Rakhmanov et al., 1995). The sphere
packing problem is equivalent to solving for s ¼ 1.
Cohn & Kumar (2007) have shown that there exist config-
urations for certain values of N which are universally optimal,
that is, globally optimal solutions for every value of s. The
known universally optimal configurations for d ¼ 3 are the
tetrahedron, the 16-cell and the 600-cell. The vertices of these
polyhedra are conjectured to be global optima for the sphere
covering problem, since their Delaunay triangulations consist
of regular spherical tetrahedra (cf. x2.5). However, for any
value of N for which a universally optimal configuration does
not exist, there is no value of s which for a configuration
minimizing EsðPÞ guarantees an optimal spherical covering.
As such, we will select a value of s on the following basis:
Kuijlaars et al. (2007) have shown that the set of points P
which minimizes EsðPÞ is well distributed when d� 1 � s< d.
We will select s ¼ 2 since longer-range potentials exhibit fewer
local minima (Wales & Ulker, 2006). We have used the PR+
conjugate gradient method (Wright & Nocedal, 1999) to find a
local minimum of EsðPÞ. The resulting configuration is a good
intermediate solution with a small covering radius.
3.2. Optimal Delaunay triangulation smoothing
Minimizing the Riesz energy of a point set reduces the
covering radius whilst considering only the relative positions
of the points. We can obtain a further reduction in covering
radius by considering the positions of a point set and the
simplices in its Delaunay triangulation. This is a well studied
problem in the computational geometry community known as
tetrahedral meshing. Given a set of points sampled from an
object (e.g. a teapot model) the objective is to move the points
in order to create a ‘nice’ Delaunay triangulation (the mesh)
whilst preserving the shape of the object. Chen (2004) defines
an optimal Delaunay triangulation (ODT) as a set of points
which minimizes the energy function
EODT ¼1
dþ 1
Xi¼1...N
Z�i
jjp� pijj2 dp; ð26Þ
where �i is the 1-ring of pi (the volume bounded by pi and its
simplicial neighbours). Minimization of this energy results in a
Delaunay triangulation whose simplices have a low circum-
radius to inradius ratio. Alliez et al. (2005) have shown that,
for a given point, the position which minimizes EODT is
p�i ¼1
Volð�iÞ
Xt2�i
VolðtÞCðtÞ; ð27Þ
where VolðtÞ and CðtÞ are, respectively, the volume and
circumcentre of simplex t. They have shown that the energy
can be minimized with guaranteed convergence by alternately
constructing the Delaunay triangulation and moving the
vertices to their optimal positions using equation (27).
For our applications the ‘object’ whose shape we must
preserve is simply S3. As such, after the optimal vertex posi-
tion has been calculated using equation (27), the vertex
position is normalized in order to bring it back onto S3. We
also calculate VolðtÞ for a spherical tetrahedron (Murakami,
2012) rather than for a Euclidean tetrahedron. Despite the
intended use for Euclidean geometries, we have found that
this method works very well in practice in S3, which is prob-
ably because of the small local curvature of S3 for large point
sets.
3.3. Local refinement
As a last step in the process of reducing the covering radius,
we use an optimization procedure to iteratively refine a
succession of local neighbourhoods. We do so by generalizing
the direct problem, by iteratively dividing B into an active set
A and a constant set C. We then minimize the maximum
circumradius of the simplices with a vertex in A. A description
of the optimization problem is given in Table 3.
Whilst the smallest active set consists of a single vertex, we
find that optimizing the vertices of one whole simplex at a time
gives better results. To do so, we alternately construct the
Delaunay triangulation and then optimize each simplex in
turn. The order in which the simplices are optimized is
determined by their circumradius, from largest to smallest.
After each update the chosen symmetry group is reapplied to
the basis set in order to maintain a consistent point set.
Since the minimization of the maximum value of a set is a
non-differentiable objective function, we use the Nelder–
Mead method (Nelder & Mead, 1965) to optimize the above
function as it is a derivative-free method. In order to avoid
dealing with the implicit constraint jpj ¼ 1 8p 2 P, we
represent the vertices using RF vectors. Representing the
vertices as RF vectors during optimization has the added
benefit of reducing the number of variables, which is parti-
cularly advantageous when using the Nelder–Mead method.
Since an RF vector representation of any 180 rotation has
infinite magnitude, we rotate the local neighbourhood under
consideration to f1; 0; 0; 0g prior to optimization, and back
again after optimization.
4. Results
Fig. 5 illustrates how each stage of the optimization process
affects the solution quality. The initial random sampling
research papers
J. Appl. Cryst. (2017). 50 Larsen and Schmidt � Improved orientation sampling for indexing 7 of 12
Table 3Model for reducing the covering radius of a local neighbourhood of apoint set.
The point set P is composed of a basis set, B, on which a chosen symmetrygroup, G, acts. The basis set, B, consists of an active set, A, which defines thelocal neighbourhood to be optimized, and a constant set, C, which contains theremaining points. The covering radius, �, is again calculated using theDelaunay triangulation, though only of the points which either are active orshare a simplicial neighbour with an active point.
Variables A ¼ fa1 2 S3; a2 2 S
3; . . .g (1)Parameters C ¼ fc1 2 S
3; c2 2 S3; . . .g (2)
G ¼ fg1 2 S3; g2 2 S
3; . . .g (3)Minimize maxf’ðtÞ j t 2 DTðPÞ ^ t \A 6¼ ;g (4)Subject to B ¼ A [ C (5)
P ¼ fb� g j b 2 B; g 2 Gg (6)
results in a distribution of simplex circumradii that is
approximately Gaussian. Minimization of the Riesz energy
significantly reduces the mean and variance of the simplex
circumradii, as well as the number of simplices. The distribu-
tion resembles a bimodal Gaussian distribution, which
suggests an ordered underlying simplex structure.
Application of ODT smoothing reduces the mean and
variance of the of simplex circumradii and results, again, in an
approximately Gaussian distribution. Lastly, the objective of
the local refinement procedure is to minimize the maximum
simplex circumradius. It can be seen that this produces a peak
around the maximum circumradius with a tail of smaller
circumradii below this.
Fig. 6 shows a comparison of our method with the methods
discussed in x1, in the range N ¼ ½960; 200 000�. We have
applied 2I60 symmetry, as it requires a small basis set and thus
allows us to quickly generate coverings of the full space of S3.
For each value of N, we have applied our method from 200
random starting configurations and taken the point set with
the lowest covering radius. It can be seen that the resulting sets
have a lower covering radius than the other methods, at both
small and large values of N.
Furthermore, our method displays a smooth decrease in
covering radius with increasing N, which is highlighted by the
almost constant covering density. We do not claim optimality
for any of our point sets; in most cases the covering radius of
the best point set was unique amongst the 200 runs. As such we
can conclude that lower covering radii could be obtained
simply by increasing the number of runs, though this is very
time consuming for large point sets.
research papers
8 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50
Figure 5Histograms showing the change in simplex circumradius at each stage in the optimization for a point set with antipodal symmetry and N = 20 000. Thehistograms show the simplex circumradius distribution after (a) initial random sampling, (b) Riesz energy minimization, (c) ODT smoothing and (d)local refinement. The maximum circumradius is reduced at every stage.
Figure 6Comparison of the covering radius (left) and the covering density (right) of random sampling from a uniform distribution with antipodal symmetry,incremental grids based on the Hopf fibration (Yershova et al., 2010), cubochoric grids (Rosca et al., 2014), b.c.c. grids with binary octahedral symmetry(Karney, 2007) and our method. For the random sampling, a mean of 105 runs was used. For the incremental grids based on the Hopf fibration, thecovering radius was calculated at every value of N in the range shown. For the covering radius optimized point sets (our method), the best result of 200runs was used.
The optimality gaps of some selected point sets generated
using our method are shown in Table 4. The gaps are below
6% at every value of N. In the Euclidean limit (N!1) the
curvature of S3 in a local area is effectively zero. For this
reason, the optimal covering in a local area should be a b.c.c.
lattice, since this is the best known covering inR3. Since a b.c.c.
lattice has a higher covering density than the simplex bound,
the optimality gaps presented here leave room for improve-
ment.
4.1. Practical application
The results presented in Fig. 6 demonstrate the evolution of
the different methods with increasing size, though all at small
sizes. For a practical pattern-indexing application, much larger
point sets are needed. Furthermore, whilst the covering radius
of a set specifies the maximum error, the distribution of errors
is also of practical interest. Fig. 7 compares the error histo-
grams of a covering-radius-optimized set and a cubochoric set,
which is used for comparison owing to its use in the widely
used EMsoft microscopy software (De Graef, 2017). In order
to generate the error histogram, 108 random orientations were
sampled; for each sampled orientation, the misorientation is
calculated to the nearest orientation in the dictionary set. A
KD-tree (Bentley, 1975) is used to quickly find the closest
dictionary orientation. In addition to a smaller maximum
error, the covering-radius-optimized set has a better overall
error distribution. This is achieved despite the use of a smaller
number of orientations.
The maximum error of the covering-radius-optimized set is
72% smaller than that of the cubochoric set. In the Euclidean
limit � / n�1=3, which suggests that a cubochoric set would
require approximately 5 times as many points to achieve the
same maximum error.
Using the symmetry relationships described in x2.4, we have
created orientation sets for every Laue group with maximum
misorientations (2�) of <1, 2, 3, 4 and 5, with optimality gaps
of less than 6% for every set. The orientation sets are available
online (Larsen & Schmidt, 2017).
5. Summary
We have shown how to construct a near-optimal sampling of
orientations. First we demonstrated that the sampling problem
is equivalent to the problem of how to distribute points on a
hypersphere. We then showed that the best measure of quality
for a point set is the covering radius, as this determines the
maximum orientational error. With the minimum covering
radius as the objective, we created sets of orientations at a
range of sizes for use in pattern indexing, and demonstrated
that the number of orientation samples needed to achieve a
desired indexing accuracy is significantly reduced as a conse-
quence.
In addition to an exact calculation of the covering radius,
which measures the quality of a set of orientations (smaller is
better), we derived a lower bound on the covering radius,
which sets a limit on the best attainable quality. The difference
between the achieved covering radius and the theoretical limit
allows us to quantify the optimality of orientation sets, which
we used to show that the sets we created are within 6% of the
optimal covering radius.
In order to use the method for indexing of diffraction
patterns, we have shown how symmetry groups can be
imposed during orientation sampling, without introducing any
research papers
J. Appl. Cryst. (2017). 50 Larsen and Schmidt � Improved orientation sampling for indexing 9 of 12
Figure 7Error histogram for a covering-radius-optimized point set with 6 106
orientations and a cubochoric set with 6:3 106 orientations. Here, bothpoint sets cover the full space of SOð3Þ, which corresponds to indexing amaterial with a triclinic crystal lattice. The covering-radius-optimizedpoint set has a lower maximum error (1.00 versus 1.72) and a betteroverall distribution of errors.
Table 4Conjectured optimality gaps for covering-radius-optimized configura-tions, with 2I60 symmetry applied.
N is the number of points in each set, � is the covering radius and �� is thecovering radius of the simplex bound on S3, conjectured to be a lower bound(Boroczky, 2004). The optimality gap percentage is 100ð�=�� � 1Þ.
N � () �* () Optimality gap (%)
8† 60.00 60.00 0.00120† 22.24 22.24 0.001920 9.05 8.73 3.683960 7.20 6.85 5.056000 6.27 5.96 5.077920 5.71 5.44 4.959960 5.27 5.04 4.6712000 4.96 4.73 4.7113920 4.72 4.50 4.7615960 4.50 4.30 4.5418000 4.33 4.13 4.7419920 4.18 4.00 4.6124000 3.93 3.76 4.7227960 3.72 3.57 4.3131920 3.56 3.41 4.3836000 3.43 3.28 4.4739960 3.31 3.17 4.6243920 3.21 3.07 4.6748000 3.11 2.98 4.4960000 2.89 2.77 4.3579920 2.63 2.51 4.4899960 2.44 2.33 4.64139920 2.19 2.09 4.83180000 2.01 1.92 4.84
† N ¼ 8 and N ¼ 120 are the point sets containing the vertices of the 16-cell and 600-cell,respectively, included here to highlight the tightness of the simplex bound for point setsconsisting of regular tetrahedral cells.
edge-effect artifacts. Using this approach we have demon-
strated how to sample from the crystallographic fundamental
zone of any of the 11 Laue groups.
Existing methods for sampling orientations have prioritized
properties such as a refinable grid-like structure, fast genera-
tion and the ability expand into spherical harmonics; we have
instead chosen to optimize the maximum error (the covering
radius) above all else. This also means that the sampling
method has very high computational requirements; the largest
point set requires approximately four days of computation
time. Nonetheless, we claim that this is a good trade-off, since
a point set must only be generated once for each desired error
level, and affords a significant performance improvement
every time a pattern is subsequently indexed.
APPENDIX ASimplex bound derivation
A1. Volume of a hyperspherical cap
The volume of a hyperspherical cap in Sd can be calculated
by projection into RF space. Since RF space is radially
symmetric about the origin, the projection of a hyperspherical
cap with radius � and centre coordinates f1; 0; 0; . . . ; 0g 2 Sd is
a sphere with radius r ¼ tanð�Þ centred at the origin. Thus, the
volume of the cap is the radial integral of the product of the
surface area of a ðd� 1Þ-sphere with the RF space density:
Cdð�Þ ¼
Ztanð�Þ
0
Sd�1ðrÞ
ð1þ r2Þ2 dr where Sd�1ð�Þ ¼
d�d=2
� d=2þ 1ð Þ�d�1:
ð28Þ
For a hyperspherical cap in S3, this gives
C3ð�Þ ¼
Ztanð�Þ
0
4�r2
ð1þ r2Þ2
dr ¼ �½2� � sinð2�Þ�: ð29Þ
This is the same as the result derived by Morawiec (2004,
2010), but without normalization.
A2. Edge length of a regular spherical tetrahedron in S3
Owing to the radial symmetry of RF space, the RF
projection of a regular spherical tetrahedron with centre
coordinates qc ¼ f1; 0; 0; 0g 2 S3 is a tetrahedron with centre
coordinates vc ¼ f0; 0; 0g and vertex coordinates
v1 ¼ fk; k; kg; v2 ¼ fk;�k;�kg;
v3 ¼ f�k; k;�kg; v4 ¼ f�k;�k; kg:ð30Þ
From this, we obtain the vertex coordinates in S3:
q1 ¼1
ð1þ 3k2Þ1=2f1; k; k; kg;
q2 ¼1
ð1þ 3k2Þ1=2f1; k;�k;�kg;
q3 ¼1
ð1þ 3k2Þ1=2f1;�k; k;�kg;
q4 ¼1
ð1þ 3k2Þ1=2f1;�k;�k; kg:
ð31Þ
The circumradius of the tetrahedron is given by the arc length
from the centre to any of the vertices:
� ¼ arccoshqc; qii ¼ arccos1
ð1þ 3k2Þ1=2
� �8i: ð32Þ
The edge length of the tetrahedron is the arc length between
any two vertices:
l ¼ arccoshqi; qji ¼ arccos1� k2
1þ 3k2
�8i 6¼ j: ð33Þ
Using equations (32) and (33) we can express the edge length
in terms of the radius:
l ¼ arccos4 cos2ð�Þ � 1
3
� �: ð34Þ
A3. Dihedral angle and solid angle of intersection
Let fq1; q2; q3; q4g be the vertices of a regular hyper-
spherical simplex in S3 with the following coordinates:
q1 ¼ 1; 0; 0; 0f g; q2 ¼ cosðlÞ;�a;�a
31=2; z
n o;
q3 ¼ cosðlÞ; a;�a
31=2; z
n o; q4 ¼ cosðlÞ; 0;�
2a
31=2; z
� ;
ð35Þ
where
a ¼1� cosðlÞ
2
� �1=2
; z ¼ sin2ðlÞ �
2
3½1� cosðlÞ�
� 1=2
: ð36Þ
When projected into RF space the tetrahedron has vertices
v1 ¼ 0; 0; 0f g; v2 ¼1
cosðlÞ�a;�
a
31=2; z
n o;
v3 ¼1
cosðlÞa;�
a
31=2; z
n o; v4 ¼
1
cosðlÞ0;�
2a
31=2; z
� :
ð37Þ
The dihedral angle of the tetrahedron is then given by
lð Þ ¼ arccosv2 v3
jv2 v3j;
v2 v4
jv2 v4j
� �¼ arccos
cosðlÞ
2 cosðlÞ þ 1
� �:
ð38Þ
Using equation (34) we can express the dihedral angle in terms
of �:
ð�Þ ¼ arccos4 cos2ð�Þ � 1
8 cos2ð�Þ þ 1
� �: ð39Þ
research papers
10 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50
The solid angle is then given by
�ð�Þ ¼ 3 ð�Þ � � ¼ 3 arccos4 cos2ð�Þ � 1
8 cos2ð�Þ þ 1
� �� �: ð40Þ
Since v1 lies at the origin, this is also the solid angle of inter-
section of a regular hyperspherical simplex and a hyper-
spherical cap placed at one of its vertices. We can verify that in
the Euclidean limit (where the curvature is zero)
lim�!0 �ð�Þ ¼ 3 arccosð1=3Þ � � ¼ arccosð23=27Þ, which is
the solid angle for a regular tetrahedron in R3, and that
�ð�=3Þ ¼ �=2, which is the solid angle of a tetrahedral cell in
the 16-cell.
APPENDIX BLaue group subset relationships
The subset relationships between the 11 Laue groups are
shown in Tables 5 and 6.
Acknowledgements
The authors thank Thomas J. Hardin for advice on spherical
and hyperspherical harmonics, Farangis Ram for discussions
on the cubochoric method, Nanna Wahlberg, Erik B. Knudsen
and Hugh Simons for proofreading and helpful suggestions on
the manuscript, and an anonymous referee for several
suggestions which have improved this work.
References
Adams, B. L., Wright, S. I. & Kunze, K. (1993). Metall. Trans. A, 24,819–831.
Alliez, P., Cohen-Steiner, D., Yvinec, M. & Desbrun, M. (2005). ACMTrans. Graph. 24, 617–625.
Altmann, S. L. (2005). Rotations, Quaternions and Double Groups.Mineola: Dover Publications.
Altschuler, E. L. & Perez-Garrido, A. (2007). Phys. Rev. E, 76,016705.
Altschuler, E. L., Williams, T. J., Ratner, E. R., Dowla, F. & Wooten,F. (1994). Phys. Rev. Lett. 72, 2671–2674.
Bentley, J. L. (1975). Commun. ACM, 18, 509–517.Boroczky, K. (1978). Acta Math. Acad. Sci. Hung. 32, 243–261.Boroczky, K. (2004). Finite Packing and Covering, Vol. 154.
Cambridge University Press.Brown, R. & Gray, A. (1967). Comment. Math. Helv. 42, 222–
236.Chen, L. (2004). Mesh Smoothing Schemes Based on Optimal
Delaunay Triangulation, Proceedings of the 13th InternationalMeshing Roundtable, pp. 109–120. http://imr.sandia.gov/papers/abstracts/Ch317.html.
Chen, Y. H., Park, S. U., Wei, D., Newstadt, G., Jackson, M. A.,Simmons, J. P., De Graef, M. & Hero, A. O. (2015). Microsc.Microanal. 21, 739–752.
Cohn, H. & Kumar, A. (2007). J. Am. Math. Soc. 20, 99–149.Conway, J. H. & Sloane, N. J. A. (1998). Sphere Packings, Lattices and
Groups. New York: Springer.Conway, J. H. & Smith, D. A. (2003). On Quaternions and Octonions.
New York: Springer.Coxeter, H., Few, L. & Rogers, C. (1959). Mathematika, 6, 147–
157.Dahms, M. & Bunge, H. J. (1989). J. Appl. Cryst. 22, 439–447.De Graef, M. (2017). EMsoft – Electron Microscopy Diffraction and
Image Computations. http://github.com/marcdegraef/EMsoft/.Emde Boas, P. van (1981). Another NP-Complete Partition Problem
and the Complexity of Computing Short Vectors in a Lattice. Report81-04. Mathematisch Instituut, Universiteit van Amsterdam, TheNetherlands.
Erber, T. & Hockney, G. (1991). J. Phys. A Math. Gen. 24, L1369–L1377.
Frank, F. (1988). Metall. Trans. A, 19, 403–408.Hardin, D. & Saff, E. (2004). Not. AMS, 51, 1186–1194.Hardin, R. H., Sloane, N. J. A. & Smith, W. D. (2017). Spherical
Codes. Book in preparation. http://neilsloane.com/coverings/.He, Y. & Jonas, J. J. (2007). J. Appl. Cryst. 40, 559–569.Karney, C. F. F. (2007). J. Mol. Graph. Model. 25, 595–604.
research papers
J. Appl. Cryst. (2017). 50 Larsen and Schmidt � Improved orientation sampling for indexing 11 of 12
Table 5Generators for the seven Laue groups which are subsets of O.
% O T D4 D2 C4 C2 C1
f1; 0; 0; 0gp p p p p p p
f0; 0; 0; 1gp p p p p p
f0; 1; 0; 0gp p p p
f0; 0; 1; 0gp p p p
f21=2=2; 0; 0; 21=2=2gp p p
f21=2=2; 0; 0;�21=2=2gp p p
f0; 21=2=2; 21=2=2; 0gp p
f0;�21=2=2; 21=2=2; 0gp p
f1=2; 1=2;�1=2; 1=2gp p
f1=2; 1=2; 1=2;�1=2gp p
f1=2; 1=2;�1=2;�1=2gp p
f1=2;�1=2;�1=2;�1=2gp p
f1=2;�1=2; 1=2; 1=2gp p
f1=2;�1=2; 1=2;�1=2gp p
f1=2;�1=2;�1=2; 1=2gp p
f1=2; 1=2; 1=2; 1=2gp p
f21=2=2; 21=2=2; 0; 0gp
f21=2=2;�21=2=2; 0; 0gp
f21=2=2; 0; 21=2=2; 0gp
f21=2=2; 0;�21=2=2; 0gp
f0; 21=2=2; 0; 21=2=2gp
f0;�21=2=2; 0; 21=2=2gp
f0; 0; 21=2=2; 21=2=2gp
f0; 0;�21=2=2; 21=2=2gp
Table 6Generators for the five Laue groups which are subsets of D6.
% D6 D3 C6 C3 C1
f1; 0; 0; 0gp p p p p
f1=2; 0; 0; 31=2=2gp p p p
f1=2; 0; 0;�31=2=2gp p p p
f0; 0; 0; 1gp p
f31=2=2; 0; 0; 1=2gp p
f31=2=2; 0; 0;�1=2gp p
f0; 1; 0; 0gp p
f0;�1=2; 31=2=2; 0gp p
f0; 1=2; 31=2=2; 0gp p
f0; 31=2=2; 1=2; 0gp
f0;�31=2=2; 1=2; 0gp
f0; 0; 1; 0gp
Kuijlaars, A. B. J., Saff, E. B. & Sun, X. (2007). J. Comput. Appl. Math.199, 172–180.
Larsen, P. M. & Schmidt, S. (2017). Orientation Sets Created viaSolution of the Hyperspherical Covering Problem, http://github.com/pmla/hyperspherical-coverings.
Li, S. F. & Suter, R. M. (2013). J. Appl. Cryst. 46, 512–524.Mason, J. K. (2009). Acta Cryst. A65, 259–266.Mason, J. K. & Schuh, C. A. (2008). Acta Mater. 56, 6141–6155.Morawiec, A. (2004). Orientations and Rotations: Computations in
Crystallographic Textures. Berlin, Heidelberg: Springer-Verlag.Morawiec, A. (2010). Acta Cryst. A66, 717–719.Morawiec, A. & Field, D. (1996). Philos. Mag. A, 73, 1113–1130.Murakami, J. (2012). Proc. Am. Math. Soc. 140, 3289–3295.Nelder, J. A. & Mead, R. (1965). Comput. J. 7, 308–313.Poulsen, H. F., Nielsen, S. F., Lauridsen, E. M., Schmidt, S., Suter,
R. M., Lienert, U., Margulies, L., Lorentzen, T. & Juul Jensen, D.(2001). J. Appl. Cryst. 34, 751–756.
Rakhmanov, E. A., Saff, E. B. & Zhou, Y. M. (1995). Comput.Methods Funct. Theory, 5, 293–309.
Ram, F., Wright, S., Singh, S. & De Graef, M. (2017). Ultramicro-scopy, 181, 17–26.
Rogers, C. A. (1958). Proc. London Math. Soc. s3–8, 609–620.Rosca, D., Morawiec, A. & De Graef, M. (2014). Modell. Simul.
Mater. Sci. Eng. 22, 075013.Schmidt, S. (2014). J. Appl. Cryst. 47, 276–284.Shoemake, K. (1992). Graphics Gems III, pp. 124–132. Academic
Press.Singh, S. & De Graef, M. (2017). Microsc. Microanal. 23, 1–10.Toth, L. F. (1964). Regular Figures. New York, Paris: Elsevier.Wales, D. J. & Ulker, S. (2006). Phys. Rev. B, 74, 212101.Wright, S. & Nocedal, J. (1999). Numerical Optimization, Vol. 2. New
York: Springer.Yershova, A., Jain, S., LaValle, S. M. & Mitchell, J. C. (2010). Int. J.
Robot. Res. 29, 801–812.
research papers
12 of 12 Larsen and Schmidt � Improved orientation sampling for indexing J. Appl. Cryst. (2017). 50